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The system ¹₁-CA₀ is known as the strongest system of the big five in reverse mathematics. It is known that some theorems represented by a ¹₂ sentence, for example Kruskal's theorem, are provable from ¹₁-CA₀ but not provable from the second strongest system ATR₀ of the big five. However, since any ¹₂ sentence is not equivalent to ¹₁-CA₀, ¹₁-CA₀ is too strong to prove such theorems. In this paper, we introduce a hierarchy dividing the set \ ¹₂: ¹₁-CA₀ \. Then, we give some characterizations of this hierarchy using some principles equivalent to ¹₁-CA₀: leftmost path principle, Ramsey's theorem for ⁰ₙ classes of N^N and determinacy for (⁰₁) ₙ classes of N^N. As an application, our hierarchy explicitly shows that the number of application of the hyperjump operator needed to prove ⁰ₙ Ramsey's theorem or (⁰₁) ₙ determinacy increases when the subscript n increases.
Suzuki et al. (Sun,) studied this question.